The rational number \(\zeta \)  as predicted, in Eq. (REF ),
from the stability criterion of closed circular orbits in the central
force problem is an interesting input in the theory. The interesting
property about this rational number is that it is a constant and so it
does not depend on the details of the orbit which one tries to
perturb. The reason for the constancy of \(\zeta \)  is the following. For
any circular orbit with radius \(r_0\)  a specific \(\zeta \)  specifies the
number of undulations of the perturbed orbit. If \(\zeta \)  is a rational
number then the number of undulations of the perturbed orbit will be
such that they form a closed geometrical structure. Now suppose one
takes another circular orbit of radius \(r_0+\delta r\)  where \(\delta r\ll r_0\) . If \(\zeta \)  has a different value on this orbit then the
number of undulations due to a perturbation will be different. In the
limit \(\delta r \rightarrow 0\)  in a continuous manner the two unperturbed
circular orbits tends to each other but the number of undulations on
the circular orbits will not match as \(\zeta \)  is not a continuous
variable but can only have discrete rational values. Consequently the
number of cycles of the perturbations will change discontinuously with
radius and the perturbed orbits cannot be closed at this
discontinuity. As we are only interested in stable, closed orbits we
can conclude that \(\zeta \)  must be a constant and not change discretely
with \(r\) . The discussion on the constancy of \(\zeta \)  as given above
closely follows the analysis given in Ref. [1]} where the
author gives a nice discussion on the role of \(\zeta \)  in the case of
non-relativistic orbits.
